#686181

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2006 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(2,5)(6,8)(9,10)(11,13)(12,15)(14,16)(17,18,22,21)(19,24,20,23)(25,26)(27,28)(29,32)(30,31);(9,11,14,12)(10,15,16,13)(17,18,22,21)(19,23,20,24)(26,31)(27,29);(17,19,22,20)(18,23,21,24)(25,27,30,29)(26,28,31,32);(1,2,4,5)(3,6,7,8)(9,12,14,11)(10,15,16,13)(17,20,22,19)(18,24,21,23)(25,28,30,32)(26,27,31,29);(1,3,4,7)(2,6,5,8)(9,13,14,15)(10,11,16,12)(17,21,22,18)(19,24,20,23)(25,29,30,27)(26,32,31,28)

Polycyclic presentation:

2^9|a^2=g,b^2=fgh,c^2=gh,d^2=fi,e^2=fi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 593377);